Complete characterization of the minimal-ABC trees

The problem of characterizing trees with minimal atom-bond-connectivity index (minimal-ABC trees) has a reputation as one of the most demanding recent open optimization problems in mathematical chemistry. Here firstly, we give an affirmative answer to the conjecture, which states that enough large minimal-ABC trees are comprised solely of a root vertex and so-called $D_z$- and $D_{z+1}$-branches. Based on the presented theoretical results here and some already known results, we obtain enough constraints to reduce the search space and solve the optimization problem, and thus, determine exactly the minimal-ABC trees of a given arbitrary order.